arXiv:1510.08624v2 [math.AP] 31 Jan 2016 · linear age-structured population dynamics; while [12] provides valuable insight into the modelling and analysis of size-structured populations,

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STRUCTURED POPULATIONS WITH DISTRIBUTED

RECRUITMENT: FROM PDE TO DELAY FORMULATION

ANGEL CALSINA, ODO DIEKMANN, AND JOZSEF Z. FARKAS

Abstract. In this work first we consider a physiologically structured populationmodel with a distributed recruitment process. That is, our model allows newly re-cruited individuals to enter the population at all possible individual states, in princi-ple. The model can be naturally formulated as a first order partial integro-differentialequation, and it has been studied extensively. In particular, it is well-posed on thebiologically relevant state space of Lebesgue integrable functions. We also formulatea delayed integral equation (renewal equation) for the distributed birth rate of the po-pulation. We aim to illustrate the connection between the partial integro-differentialand the delayed integral equation formulation of the model utilising a recent spec-tral theoretic result. In particular, we consider the equivalence of the steady stateproblems in the two different formulations, which then leads us to characterise irre-ducibility of the semigroup governing the linear partial integro-differential equation.

Furthermore, using the method of characteristics, we investigate the connection be-tween the time dependent problems. In particular, we prove that any (non-negative)solution of the delayed integral equation determines a (non-negative) solution of thepartial differential equation and vice versa. The results obtained for the particulardistributed states at birth model then lead us to present some very general results,which establish the equivalence between a general class of partial differential anddelay equation, modelling physiologically structured populations.

1. Prologue

Structured population models are of great interest both from the mathematical and theapplication point of view. Traditionally they have been formulated as partial differentialequations (PDEs for short) often with non-local (and nonlinear) boundary conditions.The early monograph [41] provided a comprehensive mathematical treatment of non-linear age-structured population dynamics; while [12] provides valuable insight into themodelling and analysis of size-structured populations, which is accessible for theoreti-cal biologists alike. We also refer the interested reader to the “green book” [37], whichprovides a systematic introduction to structured population models.

Date: February 2, 2016.1991 Mathematics Subject Classification. 92D25, 35L04, 34K30.Key words and phrases. Physiologically structured populations, distributed recruitment, delay for-

mulation, spectral theory of positive operators.

1

http://arxiv.org/abs/1510.08624v2

2 A. CALSINA, O. DIEKMANN, AND J. Z. FARKAS

Today we can safely say that the theory of semilinear equations, i.e. age-structuredmodels is well understood. The monograph [41] by G. F. Webb provides a general ma-thematical treatment of a wide class of models; see also the very useful monograph [32].In [41], among other things, the Principle of Linearised Stability (PLS for short) wasestablished for a general class of age-structured (semilinear) models. We also refer theinterested reader to the paper [28], where generators of translation semigroups were cha-racterised. The general results obtained in [28] can be applied directly to a number ofstructured population models.

On the other hand, the analysis of quasilinear models, which naturally arise whenmodelling size-structured populations, still poses difficult mathematical challenges. Re-markably, the PLS has not been established yet for fully general quasilinear PDE models.Partly because of this shortfall, a number of researchers have worked on establishing ageneral theory using a delay/integral equation formulation of physiologically structuredpopulation models, see e.g. [14, 16, 17, 20], and the references therein. As a result, acomprehensive modelling approach to build physiologically structured population modelsfrom basic principles was developed. For a class of these models, applying the sun-starcalculus for abstract integral equations, the Principle of Linearised Stability and theHopf-bifurcation theorem were established in [14, 16]. For a comprehensive overview ofthe underlying mathematical theory see e.g. [15]. We also mention, that an alternativeapproach uses integrated semigroups, see [35], and the references therein.

It is fair to say that both (delay and PDE) modelling approaches can be fruitfully ap-plied to analyse concrete problems. The reader may want to compare for example the pa-pers [19] and [26] for two different formulations (and analysis) of a classic Daphnia-algae(structured consumer-resource) model. Nevertheless, to our knowledge, the literaturelacks rigorous results establishing the connection between the two modelling approaches.One may anticipate that, at least in a simple setting, the connection is quite easily un-derstandable and can be motivated naturally, either from the mathematical, or from thebiological point of view. The ever increasing volume of the literature on structured po-pulation dynamics also underlines the need of a better understanding of the connectionbetween the two modelling approaches.

In case of a classic example, the basic linear age-structured model formulated by McK-endrick in the early 20th century, the integral equation formulation arises naturally forexample by solving the partial differential equation using the method of characteristics.The method of characteristics, to recast the partial differential equation as a pair of in-tegral equations, can be applied to more general (including nonlinear) models, too; butthe mathematical machinery often lacks a biologically inspired motivation.

In this work we are going to consider models of physiologically structured populationswith a distributed recruitment process, that is, we allow a continuum of states at birth.A vast number of models with distributed recruitment processes have been introducedmainly as models of cell populations, we just mention here some of the early works, e.g.

STRUCTURED POPULATIONS WITH DISTRIBUTED RECRUITMENT 3

[31, 37, 42], for the interested reader. More recently, we have been investigating size-structured models with a distributed recruitment process. In the papers [1, 7, 9, 10, 22,26] the emphasis was on qualitative questions, such as existence and stability of steadystates; while in [11] well-posedness of a general class of models was established. In thiswork our main goal is to establish a rigorous connection between solutions of the partialdifferential equation and of the delayed integral equation formulation of the distributedstates at birth model. Perhaps somewhat interestingly, we are going to show how afairly recent spectral theoretic result may help us to understand the connection (at leastformally) between the two formulations. With this, we also hope to ignite researchers toinvestigate the connection between the two different formulations, for different classes ofstructured population models. Any progress in this direction may not just contribute tothe general mathematical theory of structured population dynamics, but it may also shedlight on some interesting properties of the models. We just mention here as an examplethat net reproduction numbers/functions/functionals naturally arise when consideringsteady state problems (and addressing stability questions) in a PDE formulation, see e.g.[24, 26]. At the same time the net reproduction number, often denoted by R0, also arisesin the cumulative formulation as the spectral bound of the next generation operator, seee.g. [18, 33] for more details.

2. Motivation and speculations

First we are going to study a prototype linear model of a physiologically structuredpopulation with distributed recruitment, i.e. in our model individuals may enter thepopulation at all possible individual states. We refer the reader to [11], where a globalexistence result for a very general nonlinear model with distributed recruitment processwas established. Qualitative questions, such as existence and stability of steady states,and asynchronous exponential growth, were addressed in [1, 22]; while in [2] a finitedifference scheme was developed for a nonlinear distributed states at birth model to aidnumerical simulations.

Our motivation stems from the very recent papers [1, 22], in which we consideredthe positive steady state problem for a nonlinear physiologically structured populationmodel with distributed recruitment process. In the earlier paper [22], following ideasfrom [7], we treated the steady state problem of the model using semigroup methods,via analysing spectral properties of the unbounded semigroup generator. In contrast, in[1], we reformulated and studied the steady state problem in the form of a parametrisedfamily of integral equations. This then led us to arrive at the definition of a particu-lar net reproduction function (motivated by the mathematical formulation). We theninvestigated how this net reproduction function, which is naturally related to the ex-istence of positive steady states of the nonlinear model, is connected to a biologicallymeaningful and relevant net reproduction function. In the light of the results obtainedin [1, 22], it seems to be a very natural and interesting question to consider whether thetwo formulations of the positive steady state problem are equivalent. Furthermore, the


question naturally arises, whether one can establish the equivalence of the correspondingtime-dependent problems (PDE and delayed integral equation). We are going to answerthese questions in this work. With the aim of presenting results accessible to the widestpossible audiences, we start with a specific model and recall some of the important ideaspresented in [1].

The following partial differential equation may be considered as the basic physiologi-cally structured population model with distributed recruitment.

pt(s, t) + (γ(s)p(s, t))s = −µ(s)p(s, t) +

∫ 1

0

β(s, y)p(y, t) dy, s ∈ (0, 1),

γ(0)p(0, t) = 0.

(2.1)

Our model (2.1) is equipped with an initial condition p(·, 0), which determines the initialpopulation density. In this work, for the ease of presentation, we use the interval [0, 1]for the structuring variable s. The model above describes the time evolution of the size-distribution, (or the distribution of any another physiological structuring variable), of apopulation with distributed recruitment process. That is, individuals may be recruitedinto the population at different sizes/states. More precisely, the recruitment rate isdetermined by the fertility function β. As usual, γ and µ denote individual growth andmortality rates, respectively. All of the vital rates depend on the structuring variable s,in general.

We impose some natural regularity assumptions on the model ingredients: β, µ, γ.Specifically, we require γ to be strictly positive, continuously differentiable, and we as-sume that µ and β are non-negative and continuous. These conditions suffice to analysethe problem both in the framework of partial differential and integral equation. We notethat the assumption of strict positivity on γ implies that in principle there may be anumber of individuals growing beyond the maximal size. However, choosing the mortal-ity function such that it becomes sufficiently large close to the maximal size implies thatthe proportion of individuals growing beyond the maximal size is negligible. At the sametime we note that our aim here is not to handle the most general situation (i.e. to studythe model with the most relaxed smoothness assumptions on the vital rates).

For problem (2.1) the natural choice of state space is the Lebesgue space L1(0, 1). Thisis simply because convergence in the L1 norm has a clear biological interpretation, that is,it implies convergence of the density describing the size distribution of the population. Onthis state space, under the assumptions on β, µ and γ we specified earlier, the followingdensely defined operator generates a strongly continuous semigroup of bounded linearoperators.

A p = −∂

∂s(γp)− µp+

∫ 1

0

β(·, y)p(y) dy, (2.2)

where the domain of the generator A is defined as

D(A) ={p ∈W 1,1(0, 1) | γ(0)p(0) = 0

}.


It is shown (see e.g. [22]) thatA generates an eventually compact and positive semigroup.Moreover, it was shown in [22], that the semigroup is irreducible, if there exists an ε∗ > 0such that ∀ ε ∈ (0, ε∗) we have

∫ ε

0

∫ 1

1−ε

β(s, y) dy ds > 0. (2.3)

Note that this irreducibility condition has a clear biological meaning: it requires that largeindividuals produce offspring of arbitrarily small size. For example if β is continuous (asindeed we assumed), condition (2.3) holds if β(0, 1) 6= 0. Also note that condition (2.3)is sufficient, but not necessary. For example the weaker assumption, that there exists ans∗ ∈ [0, 1] and an s ∈ [s∗, 1], such that β(s∗, 1) > 0 and β(0, s) > 0 hold simultaneously,would also suffice to guarantee irreducibility of the semigroup. In fact, we are goingto return to the question of irreducibility of the semigroup, and we will formulate anecessary and sufficient condition in the next section.

Due to the eventual compactness of the governing semigroup, the spectrum of Acontains only eigenvalues of finite multiplicity, and the Spectral Mapping Theorem holdstrue. Moreover, the eigenvalues of A can be characterised implicitly via the followingfunctional equation

f(·) =

∫ 1

0

β(·, y)

γ(y)

∫ y

0

f(r) exp

{

−

∫ y

r

λ+ µ(x)

γ(x)dx

}

dr dy, (2.4)

where we introduced

f(·) =

∫ 1

0

β(·, y)p(y) dy.

More precisely, λ ∈ C is an eigenvalue of A, if and only if there exists a non-trivialf , such that (λ, f) is a solution of equation (2.4). Then, the non-trivial eigenvector pcorresponding to λ, is given by

p(·) =1

γ(·)

∫ ·

0

exp

{

−

∫ ·

r

λ+ µ(x)

γ(x)dx

}

f(r) dr. (2.5)

The sign of the spectral bound of A, (note that the spectral bound is a real and strictlydominant eigenvalue), determines the asymptotic behaviour of solutions of the linearmodel (2.1). Note that equation (2.4), which is the characteristic equation correspondingto the linear problem (2.1), is a functional equation, in contrast to the case of a singlestate at birth model, when the characteristic equation is a scalar one.

We also note that even in the case of an unbounded size-space, the asymptotic be-haviour of solutions of the linear model (or a linearised model), is determined by eigenval-ues at least if the mortality function is bounded below by a positive constant µ0. In thiscase, even though the essential spectrum of A may not be empty, it can still be containedin the half-plane λ ∈ C, Re(λ) ≤ −µ0. We refer the interested reader for example to [23]for a result of this kind, established for a size-structured population model incorporatingcannibalistic behaviour.


Equation (2.4) naturally leads us to define a bounded integral operator L as follows.

L b =

∫ 1

0

β(·, y)

γ(y)

∫ y

0

b(r) exp

{

−

∫ y

r

µ(x)

γ(x)dx

}

dr dy, (2.6)

with domain L1(0, 1). Equivalently, by introducing the function b as

b(·) =

∫ 1

0

β(·, y)p(y) dy,

and solving the equation A p = 0, one arrives at the integral equation L b = b, with Ldefined as above in (2.6). It is then shown that the operator A has spectral bound 0,(which is a simple eigenvalue if the semigroup is irreducible, e.g. when (2.3) holds), ifand only if the integral operator L has spectral radius 1. This result is in the spirit of [37,Ch.5] by Heijmans, but can be obtained directly by using a fairly recent spectral theoremfrom [40] due to Horst Thieme. We recall now this result for the reader’s convenience. Inthe theorem below s stands for the spectral bound of a linear operator, while r denotesthe spectral radius of a (bounded) linear operator. An operator O is called resolvent-positive if its resolvent set contains an interval of the form (ω,∞), and the operator(λI − O)−1 is positive for λ large enough.

Theorem 2.1. [40, Theorem 3.5] Let B be a resolvent-positive operator on X , s(B) < 0,and A = B + C a positive perturbation of B, (i.e. C is a positive linear operator withdomain including the domain of B). If A is resolvent-positive then s(A) has the samesign as r

(−C B−1

)− 1.

To apply Theorem 2.1 we split the operator A defined in (2.2) into two parts. Hence,we rewrite the partial differential equation (2.1) as the following abstract Cauchy prob-lem.

dp

dt= A p = (B + C) p, p(0) = p0, (2.7)

where we define

B p = −∂

∂s(γ p)− µp, D(B) = D(A) =

{p ∈ W 1,1(0, 1) | γ(0)p(0) = 0

}, (2.8)

C p =

∫ 1

0

β(·, y)p(y) dy, D(C) = L1(0, 1). (2.9)

Note that C : D(B) → X is positive in the sense as defined in [40]. That is, for anyp ∈ D(B) ∩ X+ we have that C p ∈ X+. Then, −CB−1 is the integral operator L definedin (2.6). This was shown in [1]. If the mortality µ is not identically zero, then it can beshown that the spectral bound of B is negative, hence the operatorsA,B and C satisfy theconditions of Theorem 2.1. That is, the connection between the two eigenvalue problems

A p = 0 p, and L b = 1 b, (2.10)

is given by Theorem 2.1. In principle, any positive eigenvector of A corresponding to theeigenvalue 0 determines a positive steady state of the PDE (2.1). Similarly, any positive


eigenvector b of L corresponding to the eigenvalue 1 will be a steady state of the delayedintegral equation, see also (2.13) below. At the same time, any positive eigenvector b ofL corresponding to the eigenvalue 1, determines a positive steady state p of the PDE(2.1) as

p(·) =1

γ(·)

∫ ·

0

b(r) exp

{

−

∫ ·

r

µ(x)

γ(x)dx

}

dr. (2.11)

In the next section we will further investigate whether we can conclude that the equi-valence of the two eigenvalue problems implies that the (positive) steady state problemsin the two different formulations are equivalent. We note however, that studying theeigenvalue problems (2.10) does not allow us to give the best possible answer for thisquestion, as we will see later in Section 5. However, this method is applicable to studyexistence of steady states of nonlinear models, see for example [1, 22]; while we anticipatethat the extension of the general results of Section 5 for nonlinear models will presentconsiderable challenges.

Our main goal in the rest of the section is to illustrate how Theorem 2.1 can lead ourintuition to recover from the PDE model (2.1) the delayed integral equation formulationof the model. Of course the problems are time-dependent, and naturally, the delayequation is formulated on a different state space. This difference is very much the samethough as for the time-independent problems (2.10). For related developments we alsorefer the interested reader to the earlier papers [18, 20].

We start by deducing the delay formulation from basic modelling principles. Similarlyto the case of a classic age or size-structured model with single state at birth, it is naturalto formulate an integral equation for the following recruitment (or birth) function

b(s, t) =

∫ 1

0

β(s, y)p(y, t) dy, s ∈ [0, 1], t > 0. (2.12)

Since γ denotes individual growth rate, i.e. the rate of change of s with respect to timet, we can compute the time τ(x, y) an individual spends to grow from size x to size y.We have

ds

dt= γ(s(t)), s(0) = x,

which yields∫ y

x

1

γ(r)dr =

∫ τ

0

1 dt = τ(x, y).

On the other hand, the probability to survive from size x to size y is given by

exp

{

−

∫ y

x

µ(r)

γ(r)dr

}

.


Hence the flux of individuals of size y at time t is

γ(y)p(y, t) =

∫ y

0

b

(

x, t−

∫ y

x

1

γ(r)dr

)

exp

{

−

∫ y

x

µ(r)

γ(r)dr

}

dx.

The formula above is meaningful at least for times t >

∫ 1

0

1

γ(s)ds =: Γ. This is because

in principle the density p, which defines b in (2.12) is not defined for negative times.Hence from (2.12) we obtain that the birth rate function b satisfies the integral equation(now with delay)

b(·, t) =

∫ 1

0

β(·, y)

γ(y)

∫ y

0

b

(

x, t−

∫ y

x

1

γ(r)dr

)

exp

{

−

∫ y

x

µ(r)

γ(r)dr

}

dxdy, (2.13)

for t > Γ.We now intend to illustrate how the integral equation (2.13) can be obtained (formally)

in the spirit of Theorem 2.1 by introducing some appropriate operators. Note that whatfollows will be done by hand-waving (on purpose, to motivate the connection to Theorem2.1), and the rigorous connection between the time-dependent problems will be given inSection 4. For a given initial condition p0, let p be the solution of the PDE (2.1). Itis natural to assume that p ∈ C

([0,∞);L1(0, 1)

). First we obtain the recruitment (as

in (2.12)) function from the population density distribution now formally, by defining amap C as follows.

C : p(⋄, ∗)︸︷︷︸

∈C([0,∞);L1(0,1))

−→ b(·, ∗)︸︷︷︸

∈C([0,∞);C(0,1))

(

=

∫ 1

0

β(·, ⋄)p(⋄, ∗) d⋄

)

. (2.14)

Note that above ⋄ stands for the size variable (first component of p) and ∗ stands for thetime variable (the second component of p). At the same time (as noted above) p can beunderstood as a continuous L1-valued function of time. Formally, C can be consideredto be the time dependent analogue of the linear operator C defined in (2.9). Next wedefine a shift operator, which is required, in contrast to the steady state problem, due tothe time-dependence. In particular, we introduce

S : b(·, ∗)︸︷︷︸

∈C([0,∞);C(0,1))

−→ b

(

·, ∗ −

∫ ⋄

·

1

γ(r)dr

)

︸︷︷︸

∈C([Γ,∞);C([0,1]2))

. (2.15)

Similarly, we define a time-dependent version of the operator −B−1 as follows.

−B−1 : b(·, ∗, ⋄)︸︷︷︸

∈C([Γ,∞);C([0,1]2))

−→ −1

γ(⋄)

∫ ⋄

0

exp

{

−

∫ ⋄

·

µ(r)

γ(r)dr

}

b(·, ∗, ⋄) d·

︸︷︷︸

∈C([Γ,∞);L1(0,1))

. (2.16)


Then, the delayed integral equation (2.13) can formally (and in an abstract sense) bewritten as

b =(−CB−1S

)b. (2.17)

Note that it is straightforward to verify that of course the integral equation (2.13) isindeed the one which arises when building the linear distributed states model followingthe steps described by Diekmann et al., see e.g. [14, 15, 20].

We would also like to point out that one would readily obtain the following (similar)delayed integral equation for the birth rate

b(·, t) =

∫ 1

0

β(·, y, P (t))

γ(y)

×

∫ y

0

b

(

s, t−

∫ t

s

1

γ(r)dr

)

exp

−

∫ y

s

µ(

r, P(

t−∫ r

s1

γ(x) dx))

γ(r)dr

ds dy,

(2.18)

for t > Γ, in the semilinear case, i.e. when µ and β depend on the total population

size, P (t) =∫ 1

0 p(s, t) ds, too. The quasilinear case, i.e. when γ depends on the totalpopulation size as well, leads to a formally more complicated (and cumbersome) delayedintegral equation. Note that, even in the case when γ depends on the total populationsize P , the partial differential equation can still be solved along characteristic lines (atleast locally), see e.g. [8, 29, 34].

3. Exploring the equivalence of the steady state problems

As we have seen in the previous section, the eigenvalue problems (2.10) are veryclosely related by Theorem 2.1. Here we further investigate the connection between thesetwo eigenvalue problems, and we also discuss the relationship between the two steadystate problems. To this end, note that if the semigroup generated by A is irreducible,then the spectral bound of A is the only eigenvalue with a positive eigenvector, see e.g.Proposition 3.5 and Theorem 3.8 in [6, C-III], as well as Theorem 1.2 and Remark 1.3 in[5]. Moreover, the (unique, normalised) eigenvector corresponding to the spectral boundis strictly positive. Similarly, if the integral operator L is irreducible, then the spectralradius of L is the only eigenvalue with a corresponding strictly positive eigenvector,see e.g. Theorem 5.2 and its corollary in [39, Ch.V]. Hence if the irreducibility of thesemigroup would imply irreducibility of the integral operator defined in (2.6) and viceversa, we could say that the two spectral problems (2.10) are equivalent. Therefore, itis very interesting to explore what is the relationship between the irreducibility of thesemigroup generated by A, and the irreducibility of the integral operator L. Note thatirreducibility of the semigroup generated by A plays an important role in the qualitativeanalysis of the nonlinear version of model (2.1), too, see e.g. [22]. At the same timewe note that the existence of a unique strictly positive eigenvector corresponding to


the eigenvalue (spectral bound) 0 of A is a sufficient, but it is clearly not a necessarycondition, for the existence of a positive steady state of problem (2.1).

We first formulate a necessary and sufficient condition for the irreducibility of thesemigroup generated by A, for a continuous and non-negative β. Let us recall from [6,C-III] that a positive semigroup T on the Banach lattice L1(0, 1) is said to be irreducible,if for any given f ∈ L1

+(0, 1) and φ ∈ L∞+ (0, 1), we have 〈T (t∗) f, φ〉 > 0 for some t∗ ≥ 0,

where 〈·, ·〉 stands for the natural pairing between L1 and its dual L∞.We also recall another equivalent characterisation of irreducibility (see again [6, C-

III]). In particular, the semigroup T is irreducible if it does not admit invariant closedideals other than the trivial ones. We also note that in L1(0, 1) the non-trivial idealsare characterised by equivalence classes of functions vanishing on a measurable subset of(0, 1) with positive measure, see e.g. [6]. Our main result is formulated in the followingtheorem.

Theorem 3.2. The semigroup T generated by the operator A defined in (2.2) is irre-ducible, if and only if

∀α ∈ (0, 1), ∃ (s∗, y∗) ∈ [0, α]× [α, 1], such that β(s∗, y∗) > 0. (3.19)

Proof. Note that is not too difficult to see that the first definition of irreducibility,which we recalled earlier, is equivalent to the following condition: for every 0 6≡ p0 ∈L1+(0, 1) one has that

⋃

t≥0

ess supp (T (t) p0) = [0, 1]. (3.20)

First to see the necessity of condition (3.19), note that if there was an α ∈ (0, 1), suchthat for every (s, y) ∈ [0, α] × [α, 1], β(s, y) = 0 would hold; then the ideal consistingof functions vanishing almost everywhere on the interval (0, α) would be invariant underthe semigroup T .

To prove that condition (3.19) is also sufficient, we note that for any initial condition0 6≡ p0 ∈ L1

+(0, 1), if for some t ≥ 0, s∗ ∈ ess supp(T (t) p0

), then ∀ s > s∗, ∃ t ≥ t,

such that s ∈ ess supp (T (t) p0). Note that, this in particular implies that for any0 6≡ p0 ∈ L1

+(0, 1) we have

⋃

t≥0

ess supp (T (t) p0) = [smin, 1],

for some smin ∈ [0, 1). We are going to prove that in fact smin > 0 is not possible. Tothis end, we define a function R as follows.

R(x) = inf {s ∈ [0, 1) |β(s, τ) > 0, for some τ ≥ x} , x ∈ [0, 1). (3.21)

First note that assumption (3.19) and the continuity of β imply that R(0) = 0, and thatR(x) < x holds, for every x ∈ (0, 1).


For any given initial condition p0 ∈ L1+(0, 1), starting with any x0 ∈ (0, 1) from the

essential support of p0, we define a sequence as follows.

xn+1 := R(xn), n ≥ 0.

Note that, ∀n ≥ 0, xn ∈ ess supp (T (t) p0), for some t ∈ [0, nΓ], hence xn ≥ smin, ∀n ∈N. Since the sequence xn is decreasing (and bounded), it is convergent. The only fixed-point of R is 0. Hence, to show that xn tends to 0 we will prove that R is right-continuouson [0, 1).

Assume by the way of contradiction that R is not right-continuous at some pointx∗ ∈ [0, 1). That is, ∃ ε > 0 such that R

(x∗ +

1n

)−R(x∗) > ε, ∀n ∈ N, and n > n∗ such

that x∗ +1n< 1.

Then, it follows from the definition of R, that ∃ s∗ ∈(R(x∗), R

(x∗ +

1n

)), and τ ≥ x∗

such that β(s∗, τ) > 0 holds, ∀n > n∗.Since β is continuous on the square [0, 1] × [0, 1], ∃n′ > n∗ such that we have

β(s∗, τ +

1n′

)> 0. This implies R

(x∗ +

1n′

)≤ s∗, a contradiction.

Therefore, since xn is decreasing, we have

limn→∞

xn+1 = limn→∞

R(xn) = R(

limn→∞

xn

)

,

that is, limn→∞

xn is the fixed point 0 of R. Therefore, it follows that smin = 0, hence

(3.20) holds, and the proof is completed. 2

Remark 3.3 Note that the definition of the function R in (3.21) can be clearly moti-vated from the biological point of view. Indeed, for any size s ∈ (0, 1), the value R(s)is the smallest possible size of offspring produced by individuals of size s in the rest oftheir lifetime.

Next we study the irreducibility of the integral operator L defined in (2.6). First wenote that it follows from [39, Ch.V] that L is irreducible, if and only if for every setI ⊂ [0, 1] of positive Lebesgue measure, we have

∫

[0,1]\I

∫

I

(∫ 1

y

β(s, r)

γ(r)exp

{

−

∫ r

y

µ(σ)

γ(σ)dσ

}

dr

)

dy ds > 0. (3.22)

If there was an s∗ ∈ (0, 1), such that β(s, y) = 0, ∀ (s, y) ∈ [0, s∗] × [s∗, 1], then withI = [s∗, 1], condition (3.22) would not hold. Hence we conclude that criterion (3.19) isnecessary for the irreducibility of the integral operator L.

On the other hand, criterion (3.19) is clearly not sufficient for the irreducibility of theintegral operator L. For example, any continuous and non-negative β, strictly positiveon the open rectangle (0, 0.1) × (0, 1), but vanishing on [0.1, 1] × [0, 1], would satisfycondition (3.19), but clearly not condition (3.22).


On the other hand, since the kernel function in the double integral in (3.22) is conti-nuous on the square (0, 1)× (0, 1), it is clear that the condition

∫ 1

s

β(s, r)

γ(r)exp

{

−

∫ r

s

µ(σ)

γ(σ)dσ

}

dr > 0, ∀ s ∈ [0, 1), (3.23)

does imply condition (3.22). Moreover, since the function 1γ(·) exp

{

−∫ ·

⋄µ(σ)γ(σ) dσ

}

is

strictly positive, condition (3.23) is equivalent to∫ 1

s

β(s, r) dr > 0, ∀ s ∈ [0, 1). (3.24)

This (sufficient) condition of irreducibility of L also has a clear biological interpretation.It requires that offspring of any size s is ”produced” by individuals of some larger size.

We summarize formally our findings on irreducibility in the following corollary.

Corollary 3.4. Irreducibility of the integral operator L implies irreducibility of the semi-group T generated by the operator A, but not vice versa.

In the light of the results above, let us give (for now) a partial answer for the question ofequivalence of the PDE formulation (2.1), and the delayed integral equation formulation(2.13) (see also (4.26) below), at a steady state.

(1) If both the semigroup generated by A and the integral operator L are irreducible,for example if condition (3.24) holds, then there are unique strictly positive(normalised) vectors p and b satisfying (2.10), and by Theorem 2.1 the steadystate problems in the two different formulations are equivalent.

(2) If the semigroup generated by A is irreducible, but L is not irreducible, thenthere is still a unique strictly positive (normalised) vector p satisfying the firstequation in (2.10). This p then determines a unique non-negative (but perhapsnot strictly positive) b satisfying the second equation in (2.10) as

b(·) =

∫ 1

0

β(·, y)p(y) dy. (3.25)

In fact formula (2.11) shows that even though L may not be irreducible it doesnot have another non-negative eigenvector corresponding to its spectral radius 1.Hence once again we may conclude that the steady state problems are equivalent.

(3) In the case when neither the semigroup generated by A, nor the integral operatorL are irreducible, the geometric multiplicities of the spectral bound 0 of A, andthe spectral radius 1 of L may be greater than one, and may not be equal toeach other. In this case in principle it is possible that two different non-negative(normalised) eigenvectors of A corresponding to the spectral bound 0 determinethe same non-negative eigenvector of L corresponding to its spectral radius 1,via formula (3.25). Hence we cannot conclude that the steady state problems inthe two different formulations are equivalent.


Remark 3.5 Note that, as we will see later in Section 5, we will establish the equi-valence of the steady state problems in the two different formulations, using a differentapproach. However, the approach and the results we established here are actually appli-cable to the nonlinear version of model (2.1). That is, when the vital rates depend onthe total population size P . In that case the only (formal) difference is that at the steadystate we deal with families of operators AP and LP parametrised by the total populationsize P , see e.g. [1, 22] for more details. At the same time we note that the extension ofthe results of Section 5 to nonlinear (in particular quasilinear) models, promises to be achallenging problem.

4. On the equivalence of the time dependent problems

Next we study the connection between the PDE and the delayed integral equationformulation of the linear distributed states at birth model, via establishing the connectionbetween solutions of the two models. Let us start by recalling from [16] the formaldefinition of the initial value problem for the delay equation (2.13) on the state spaceL1([−Γ,∞);L1(0, 1)

), with initial condition φ ∈ L1

([−Γ, 0];L1(0, 1)

).

b(·, t) =

∫ 1

0

β(·, y)

γ(y)

∫ y

0

b

(

x, t−

∫ y

x

1

γ(r)dr

)

exp

{

−

∫ y

x

µ(r)

γ(r)dr

}

dxdy, t > 0,

b(·, t) =φ(·, t), t ∈ [−Γ, 0]. (4.26)

Let us also recall from [16] the definition of a solution of the delay equation (4.26). Fora given φ ∈ L1

([−Γ, 0];L1(0, 1)

), b ∈ L1

loc

([−Γ,∞);L1(0, 1)

)is a solution of the delay

equation (2.13) if b(·, t) = φ(·, t) for t ∈ [−Γ, 0], and b satisfies (4.26) for t > 0.We formulate our main results concerning the connection between solutions of the

initial value problem for the linear partial differential equation (2.1) and the delay equa-tion (4.26) via two theorems. To establish the connection between (2.1) and (4.26) weconsider an auxiliary problem, namely, the following inhomogeneous linear initial valueproblem.

dp

dt= B p+ f, t > 0, p(0) = p0, (4.27)

where B is defined in (2.8), and f ∈ L1((0,∞);L1(0, 1)

). Recall, for example from [38],

the definition of a mild solution p of (4.27). If B is the generator of a C0 semigroup Ton L1(0, 1), then for p0 ∈ L1(0, 1), the function p given as

p(t) = T (t) p0 +

∫ t

0

T (t− τ)f(τ) dτ, 0 ≤ t, (4.28)

is called the mild solution of (4.27).Let us also recall a usual definition when dealing with delay equations. Namely, for a

function f ∈ L1([−Γ,∞);L1(0, 1)) and for t ≥ 0, we define ft ∈ L1([−Γ, 0];L1(0, 1)) asft(τ) = f(t+ τ), τ ∈ [−Γ, 0]. In particular, notice that f0 = f |[−Γ,0].


Now we also define a linear operator K from L1([−Γ, 0];L1(0, 1)) to L1(0, 1).

(Ku) (·) =1

γ(·)

∫ ·

0

u

(

x,−

∫ ·

x

1

γ(r)dr

)

exp

{

−

∫ ·

x

µ(r)

γ(r)dr

}

dx. (4.29)

Our main tool to establish the connection between solutions of the delay and the partialdifferential equation is the following lemma.

Lemma 4.6. Let us assume that b ∈ L1([−Γ,∞), L1(0, 1)). Then the function t →K bt, t ∈ [0,∞), is the mild solution of the inhomogeneous linear initial value problem

dp

dt= B p+ b, t > 0, p(0) = K b0. (4.30)

Proof. As we can see from (4.28), the mild solution depends on the semigroup Tgenerated by B. Hence we first need to determine explicitly the semigroup T . To thisend we consider the initial value problem (4.27) with f ≡ 0. In fact, to obtain thesemigroup T , we solve the corresponding homogeneous partial differential equation

pt(s, t) + (γ(s)p(s, t))s = −µ(s)p(s, t), s ∈ (0, 1),

γ(0)p(0, t) = 0,(4.31)

by integrating along characteristic curves. The characteristic curves are determined bythe following system of ordinary differential equations.

s(t) = γ(s(t)), p(t) = −(µ(s(t)) + γ′(s(t)))p(t), s(0) = s0, p(0) = p0(s0). (4.32)

The solution of system (4.32) is given by(

S(t; s0

), p0(s0)exp

{

−

∫ t

0

(µ(S(τ ; s0)

)+ γ′

(S(τ ; s0)

))dτ

})

, (4.33)

where the function S is implicitly defined via

∫ S(t;s0)

s0

1

γ(r)dr = t. (4.34)

Note that S(t; s0

)is the size of individuals at time t, who were of size s0 at time 0.

Hence the solution of the homogeneous partial differential equation (4.31) is given by

p(s, t) = p0(S0(t; s)) exp

{

−

∫ t

0

(µ(S(τ ;S0(t; s)

))+ γ′

(S(τ ;S0(t; s)

)))dτ

}

, (4.35)

for t ≤

∫ s

0

1

γ(r)dr, and p(s, t) = 0 otherwise; where the function S0 is implicitly defined

via∫ s

S0(t;s)

1

γ(r)dr = t.


Note that S0(t; s) is the size of an individual at time 0, who is of size s at time t.Introducing the new integration variable r = S(τ ;S0(t; s)) in (4.35) yields the followingformula for the semigroup T .

(T (t)p)(s) =

0, t >

∫ s

0

1

γ(r)dr

p(S0(t; s)

) γ(S0(t;s))γ(s) exp

{

−

∫ s

S0(t;s)

µ(r)

γ(r)dr

}

, t ≤

∫ s

0

1

γ(r)dr

.

(4.36)Next we show that indeed B is the generator of the semigroup T given in (4.36). To thisend we prove that for any p ∈ D(B), the right-hand derivative of T (t) p at t = 0 is indeedB. That is,

limt→0+

∫ 1

0

∣∣∣∣

(T (t)p)(s) − p(s)

t− (Bp)(s)

∣∣∣∣ds = 0, p ∈ D(B). (4.37)

On the one hand, utilising that the condition t >

∫ s

0

1

γ(r)dr is equivalent to s < S(t; 0),

we have

∫ S(t;0)

0

∣∣∣∣

(T (t)p)(s) − p(s)

t− (Bp)(s)

∣∣∣∣ds =

∫ S(t;0)

0

∣∣∣∣−p(s)

t− (Bp)(s)

∣∣∣∣ds (4.38)

≤1

t

∫ S(t;0)

0

|p(s)| ds+

∫ S(t;0)

0

|(Bp)(s)| ds =S(t; 0)

t|p(ξ)|+

∫ S(t;0)

0

|(Bp)(s)| ds,

(4.39)

for some ξ ∈ [0, S(t; 0)] (recall that W 1,1 functions are continuous). From (4.34) we have

that S(t;0)t

≤ maxr∈[0,1]

γ(r), and since p ∈ D(B), both terms in the right-hand side of (4.39)

tend to 0 as t→ 0+.


On the other hand from (4.36) we have

p(S0(t; s)

) γ(S0(t;s))γ(s) exp

{

−

∫ s

S0(t;s)

µ(r)

γ(r)dr

}

− p(s)

t− (Bp)(s)

=

p(S0(t; s)

)exp

{

−

∫ s

S0(t;s)

µ(r) + γ′(r)

γ(r)dr

}

− p(s)

t− (Bp)(s)

= exp

{

−

∫ s

S0(t;s)

µ(r) + γ′(r)

γ(r)dr

}

p(S0(t; s))− p(s)

S0(t; s)− s

S0(t; s)− s

t+ γ(s)p′(s)

+ p(s)

exp

{

−∫ s

S0(t;s)µ(r)+γ′(r)

γ(r) dr}

− 1

t+ µ(s) + γ′(s)

→ 0, (4.40)

as t→ 0+, for almost every s ∈[S0(t; 0), 1

]. This is because p ∈ D(B), and therefore the

pointwise derivative of p coincides with its distributional derivative almost everywhere(see for instance Theorem 1 in [36, Sect. 1.1.3]), and

∂

∂tS0(t; s)

∣∣∣∣t=0

= −γ(s).

Hence we have∫ 1

S(t;0)

∣∣∣∣

(T (t)p)(s) − p(s)

t− (Bp)(s)

∣∣∣∣ds → 0, (4.41)

as t → 0+, by the Lebesgue dominated convergence theorem; in particular, since thedifference quotients of a W 1,1 function are bounded uniformly by the L1 norm of itsderivative.

We now compute the mild solution of (4.27) by means of the variation of constantsformula (4.28), with inhomogeneity and initial condition given by (4.30). Using thesemigroup T given in (4.36) we readily have, writing b(s, τ) for b(τ)(s),

(T (t)p0)(s)

=

0, t >

∫ s

0

1

γ(r)dr

1γ(s)

∫ S0(t;s)

0

b

(

σ,−

∫ S0(t;s)

σ

1

γ(r)dr

)

exp

{

−

∫ s

σ

µ(r)

γ(r)dr

}

dσ, t ≤

∫ s

0

1

γ(r)dr

.

(4.42)


Similarly, we have

(T (t− τ)b(τ)) (s)

=

0, t− τ >

∫ s

0

1

γ(r)dr

b(S0(t− τ ; s), τ

) γ(S0(t−τ ;s))γ(s) exp

{

−

∫ s

S0(t−τ ;s)

µ(r)

γ(r)dr

}

, t− τ ≤

∫ s

0

1

γ(r)dr

.

(4.43)

From (4.43) we have∫ t

0

(T (t− τ)b(τ)) (s) dτ

=

t∫

max

{

0, t−

∫ s

0

1

γ(r)dr

}

b(S0(t− τ ; s), τ

) γ(S0(t− τ ; s)

)

γ(s)exp

{

−

∫ s

S0(t−τ ;s)

µ(r)

γ(r)dr

}

dτ

=1

γ(s)

s∫

max{S0(t; s), 0

}

b

(

σ, t −

∫ s

σ

1

γ(r)dr

)

exp

{

−

∫ s

σ

µ(r)

γ(r)dr

}

dσ, (4.44)

where to obtain the last equality we introduced the new variable σ := S0(t− τ ; s).Adding (4.42) and (4.44) together, and using

−

∫ S0(t;s)

σ

1

γ(r)dr = t−

∫ s

σ

1

γ(r)dr,

we have

p(t) := T (t)p0 +

∫ t

0

T (t− τ)b(τ) dτ = Kbt, t ∈ [0,∞). (4.45)

That is, the function K bt is the mild solution of the inhomogeneous initial value problem(4.30). 2

Now we are in the position to formulate our first main result regarding the connectionof solutions of the partial differential equation (2.1) and the delay equation (4.26).

Theorem 4.7. For a given φ, if b is the unique non-negative solution of (4.26), thenthe (unique) mild solution p of (4.27) with

f = b|t∈(0,∞), p0(·) =1

γ(·)

∫ ·

0

b

(

x,−

∫ ·

x

1

γ(r)dr

)

exp

{

−

∫ ·

x

µ(r)

γ(r)dr

}

dx, (4.46)

i.e., the function given by formula (4.28), is also the (unique) mild (i.e. the semigroup)solution of (2.1) with initial condition p0.


Proof. Using the definitions of the linear operators C in (2.9), and K in (4.29), forthe solution b(t) := b(·, t) of (4.26) we have by hypothesis that b(t) = CK bt. On theother hand, since p0 = K b0, from Lemma 4.6, it follows that p(t) = K bt. So,

T (t) p0 +

∫ t

0

T (t− τ) C(p(τ)) dτ = T (t) p0 +

∫ t

0

T (t− τ) CK bτ dτ

= T (t) p0 +

∫ t

0

T (t− τ) b(τ) dτ = p(t), (4.47)

where the last equality is just the definition of the mild solution of (4.27). Thus, p(t)is the unique solution of the variation of constants equation corresponding to problem(2.1). Hence p(t) is the (semigroup) mild solution of problem (2.1) (see also its abstractformulation (2.7)), see for example [38, Sect.3.1]. 2

Our second result is formulated in the following theorem.

Theorem 4.8. Let us assume that p0 ∈ L1+(0, 1) is such that the functional equation

p0(·) =1

γ(·)

∫ ·

0

φ

(

x,−

∫ ·

x

1

γ(r)dr

)

exp

{

−

∫ ·

x

µ(r)

γ(r)dr

}

dx, (4.48)

has a non-negative solution φ ∈ L1 ((0, 1)× [−Γ, 0]). Then, the unique non-negative mildsolution p of (2.1) with p(·, 0) = p0(·) determines a non-negative solution b of (4.26),given by

b(·, t) =

∫ 1

0

β(·, y)p(y, t) dy, t > 0; b(·, t) = φ(·, t), t ∈ [−Γ, 0]. (4.49)

Proof. The mild solution p of (2.1) satisfies the variation of constants equation

p(t) = T (t) p0 +

∫ t

0

T (t− τ) (Cp) (τ) dτ, (4.50)

hence it is also a mild solution of the inhomogeneous equation (4.27) (with the sameinitial condition), with f = C p. Now we define b as in (4.49), i.e., b(t) = C(p(t)) fort > 0, and b(t) = φ(t) for t ∈ [−Γ, 0]. It turns out then, that p solves (4.30) in the mildsense (note that p0 = Kφ = K b0 by (4.48)). Hence, by Lemma 4.6,

p(t) = K bt, t ≥ 0, (4.51)

holds. From equation (4.51) we have now, for t > 0,

b(t) = C(p(t)) = C(K bt), (4.52)

and the proof is completed. 2

The previous two theorems allow us to draw the following somewhat interesting con-clusion.

Corollary 4.9. Assume that the delay equation (4.26) has two solutions b1 and b2, suchthat φ1 = b1|t∈[Γ,0] and φ2 = b2|t∈[Γ,0] determine the same function p0 via the right-handside of equation (4.48). Then b1(·, t) ≡ b2(·, t), ∀t > 0 holds.


We note that it can be shown that for any non-negative integrable function φ, theright-hand side of (4.48) determines a non-negative integrable function p0.

Intuitively it is also clear, that for any given initial size-distribution p0 we can calculatewhat was the size of an individual, which at time 0 is of size y, at any given timet ∈ [−Γ, 0], (but note that we need to make sure that the size remains positive). Thatis, it is shown that for any initial density p0 ∈ L1

+(0, 1), equation (4.48) admits a non-negative solution φ ∈ L1 ((0, 1)× [−Γ, 0]). We show this here for the special case ofγ ≡ 1 and µ ≡ 0. For any given size-distribution p0 ∈ L1

+(0, 1) let us choose an arbitraryage-size-distribution P ∈ L1

+ ((0, 1)× (0, 1)), such that for any size y ∈ (0, 1) we have

p0(y) =

∫ 1

0

P (y, a) da. (4.53)

For example, we can choose

P (y, a) =p0(y)

yχ[0,y](a), y, a ∈ (0, 1).

Further, we define φ ∈ L1+ ((0, 1)× (−1, 0)) as

φ(s, t) :=

{P (s− t,−t) if s ≤ t+ 1

0 if s > t+ 1

}

. (4.54)

We then have∫ y

0

φ(x, x−y) dx =

∫ y

0

P (y, y−x) dx =

∫ y

0

P (y, a) da =

∫ 1

0

P (y, a) da = p0(y), (4.55)

that is (4.48) holds. Also we have∫ 0

−1

∫ 1

0

φ(s, t) ds dt

=

∫ 0

−1

∫ t+1

0

P (s− t,−t) ds dt =

∫ 1

0

∫ y

0

P (y, a) da dy =

∫ 1

0

p0(y) dy <∞, (4.56)

hence φ ∈ L1+ ((0, 1)× (−1, 0)) indeed, as required. The existence of an appropriate φ

for a given p0 in the general case (i.e. when γ 6≡ 1 and µ 6≡ 0) is shown similarly, but theformulas are much more cumbersome.

5. General results

Motivated by the results of the previous sections, in this section we establish theequivalence between a general class of PDE’s and their corresponding delay (renewal)equation formulations. As we will see, the PDE model (2.1) discussed in the previoussections fits into the general framework we present here. However, in contrast to theprevious section, we take a slightly different point of view. To be able to work with greatergenerality, our starting point is the semigroup T0 generated by an operator B, whichis assumed to describe individual development (e.g. individual growth) and survival.


Note that the operator defined in (2.8) is a particular example of such an operator B(that is why purposefully we use the same notation throughout the section). Here, toavoid cumbersome formulas and calculations, we assume that time is scaled such that themaximal age an individual may attain equals 1. This then amounts to that the semigroupT0 is nilpotent, in particular T0(t)u = 0 holds, for t ≥ 1, for any u ∈ X . We assume thatin general reproduction (or recruitment of individuals) is described by a bounded linearoperator C : X → X . Again, the operator defined in (2.9) is a particular example ofsuch a general recruitment operator C.

Once the semigroup T0 is known (given), the semigroup T generated by A = B + Ccan be constructed by solving pointwise the variation of constants equation

T (t) = T0(t) +

∫ t

0

T0(t− s) C T (s) ds. (5.57)

Applying C to (5.57) we obtain

L(t) = L0(t) +

∫ t

0

L0(t− s)L(s) ds, (5.58)

where we definedL0(t) := C T0(t), and L(t) := C T (t).

The solution of equation (5.58) is obtained by generation expansion as

L =

∞∑

n=1

Ln⊗,

see Section 2 of [20] for more details. Once L is known, equation (5.57) becomes anexplicit formula for the semigroup T ,

T (t) = T0(t) +

∫ t

0

T0(t− s)L(s) ds. (5.59)

Lemma 5.10. We have

(λ I − A)−1 = (λ I − B)−1(

I − L0(λ))−1

, (5.60)

where L0(λ) stands for the Laplace transform of L(t).

Proof. Taking the Laplace transform of (5.58) we have

L(λ) =(

I − L0(λ))−1

L0(λ). (5.61)

Laplace transforming (5.59), and using that the Laplace transform of a strongly conti-nuous semigroup coincides with the resolvent operator of its generator (see for example[21, Sect.II]), yields

(λ I − A)−1 = (λ I − B)−1(

I + L(λ))

. (5.62)

Combining formulas (5.61) and (5.62) yields (5.60). 2


Corollary 5.11.

σ(A) ={

λ ∈ C | 1 ∈ σ(

L0(λ))}

. (5.63)

Proof. Since T0 is nilpotent, its resolvent (λ I − B)−1 is analytic on C. 2

Let us now turn to the delay formulation of the abstract Cauchy problem

dp

dt= A p, p(0) = p0. (5.64)

By this we mean the renewal equation (RE)

b(t) =

∫ 1

0

L0(a) b(t− a) da, (5.65)

together with the initial-history condition

b(θ) = φ(θ), −1 ≤ θ ≤ 0, (5.66)

with φ ∈ L1 ([−1, 0];X ) =: Y. Note that the delay equation (4.26) becomes a specialcase of (5.65)-(5.66). That is, (5.65)-(5.66) coincides with (4.26), when the operators Band C are defined as in (2.8), (2.9).

The solution of (5.65)-(5.66) is given by

b(t) = f(t) +

∫ t

0

L(a)f(t− a) da, t > 0, (5.67)

where

f(t) = f(t, φ) :=

∫ 1

t

L0(a)φ(t− a) da, (5.68)

with the understanding that the integral above in (5.68) equals zero for t ≥ 1.The definition

(S(t)φ) (θ) = b(t+ θ;φ) = bt(θ;φ) (5.69)

yields a strongly continuous semigroup on Y. A straightforward Laplace transform cal-culation shows that

(λI − A)−1φ(θ) = eλθc+

∫ 0

θ

eλ(θ−σ)φ(σ) dσ, (5.70)

where A is the generator of S, and

c =(

I − L0(λ))−1

∫ 1

0

L0(a)

∫ 0

−a

e−λ(a+σ)φ(σ) dσ da. (5.71)

(5.70)-(5.71) then shows that the spectra of A and A coincide, which already indicatesa strong connection between the semigroups S and T . We now further explore thisconnection.

We define the linear operator K : Y → X as

K φ =

∫ 0

−1

T0(−s)φ(s) ds =

∫ 1

0

T0(a)φ(−a) da. (5.72)


Note that the operator K defined in (4.29) is a special case of K defined above in (5.72).

Lemma 5.12.

b(t;φ) = C T (t) (K φ). (5.73)

Proof. Recall (5.68), and recall that T0 = 0 for t ≥ 1. It follows that

f(t;φ) =

∫ 1+t

t

L0(a)φ(t− a) da =

∫ 0

−1

L0(t− σ)φ(σ) dσ

= L0(t)

∫ 0

−1

T0(−σ)φ(σ) dσ = L0(t) (K φ), (5.74)

where we used that L0(t− σ) = C T0(t− σ) = C T0(t) T0(−σ) = L0(t) T0(−σ).Inserting the expression for f into (5.67) we obtain

b(t;φ) = L0(t)K φ+

∫ t

0

L(a)L0(t− a) (K φ) da. (5.75)

Similarly to (5.57)-(5.58), recalling T (t) = T0(t)+∫ t

0T (s) C T0(t− s) ds, and its counter-

part L(t) = L0(t) +∫ t

0L(s)L0(t− s) ds , we have

b(t;φ) = L(t)K φ = C T (t) (K φ). (5.76)

2

Note the similarity between Lemma 4.6 and Lemma 5.12 above. The connectionbetween the abstract Cauchy problem (5.64) and the renewal equation (5.65)-(5.66) isestablished in the following theorem.

Theorem 5.13.

T (t) (K φ) = K S(t)φ. (5.77)

Proof. We have

T (t) (K φ) = T0(t) (K φ) +

∫ t

0

T0(t− s)L(s) (K φ) ds.

Since

T0(t) (K φ) =

∫ 0

−1

T0(t− s)φ(s) ds, and L(s) (K φ) = b(s), (5.78)

we have

T (t) (K φ) =

∫ t

−1

T0(t− s) b(s) ds =

∫ t

t−1

T0(t− s) b(s) ds

=

∫ 0

−1

T0(−σ) b(t+ σ) dσ = K bt = K S(t)φ. (5.79)

2

Again, note the similarity between Theorems 4.7-4.8 and Theorem 5.13.In the lemma below R(O) stands for the range of an operator O.


Lemma 5.14. For t ≥ 1 the inclusion R(T (t)) ⊂ R(K) holds.

Proof. For t ≥ 1 the identity (5.59) reduces to

T (t) =

∫ t

0

T0(t− s)L(s) ds,

and∫ t

0

T0(t− s)L(s) ds =

∫ t

t−1

T0(t− s)L(s) ds =

∫ 0

−1

T0(−σ)L(t+ σ) dσ

⇒ ∀ψ ∈ X , T (t)ψ = K

θ 7→ L(t+ θ)ψ︸︷︷︸

φ

, for t ≥ 1.

2

Remark 5.15 In view of Lemma 5.14, Theorem 5.13 tells us that the large timebehaviour of T can be described in terms of the large time behaviour of S.

The equivalence of the steady state problems in the two different formulations isestablished in the following theorem. Below p and φ stand for time-independent solutionsof (5.64) and (5.65)-(5.66), respectively.

Theorem 5.16. If S(t)φ = φ, then T (t) p = p, where p = K φ. On the other hand, ifT (t) p = p, then S(t)φ = φ, with φ = C p, i.e. φ(θ) = C p, −1 ≤ θ ≤ 0.

Proof. First note that if φ is such that S(t)φ = φ holds, then

T (t) p = T (t)K φ = K S(t)φ = K φ = p,

shows that p = K φ is a steady state of (5.64).On the other hand, for a time-independent solution p of (5.64), for t ≥ 1 we have

p = T (t) p = T0(t) p+

∫ t

0

T0(t− s) C pds

=

∫ t

t−1

T0(t− s) ds C p =

∫ 1

0

T0(a) da C p.

Applying C to the identity above we obtain

C p = C p

∫ 1

0

L0(a) da, (5.80)

which means that C p is a constant solution of the renewal equation (5.65)-(5.66), andthe proof is completed. 2


6. Concluding remarks

In this work first we considered a structured population model with distributed statesat birth. The model can naturally be formulated as a first order partial differential equa-tion on the appropriate state space of Lebesgue integrable functions. At the same timewe note that the model belongs to a general class of physiologically structured popula-tion models discussed for example in [20]. Our main goal was to investigate how one canrecover the corresponding delay (renewal) equation formulation of the model, similarlyto the Volterra integral equation formulation of a model with single state at birth. In-terestingly enough, the connection between the two formulations can be motivated (atleast in our opinion) by a fairly recent spectral theoretic result in [40], due to H. Thieme.In particular, as we have seen in Section 2, Theorem 2.1 provides a rigorous connec-tion between two spectral problems arising from the different formulations at the steadystate, at least when some appropriately defined operators are irreducible. In general,irreducibility plays an important role when transforming steady state problems (also ofnonlinear models) into spectral problems. It guarantees the existence and uniquenessof a (normalised) strictly positive eigenvector corresponding to the spectral bound (orradius) of an operator. Partly for this reason, we explored the connection between theirreducibility of the semigroup and the integral operator, arising from the two differentformulations of the steady state problem. We showed that irreducibility of the integraloperator implies irreducibility of the semigroup, but not vice versa. We also note thatTheorem 2.1 already shows that the PDE and delay formulations are “asymptoticallyequivalent”. That is, in the generic case (when the spectral bound of A is not zero)solutions of the PDE and the delay equation grow or decay simultaneously.

In Section 4 we established the connection between the two time-dependent problemsvia Theorems 4.7 and 4.8. It turns out, that for any solution of the partial differentialequation there exists a corresponding solution of the delayed integral equation and viceversa. As far as we know the results presented here are the first ones comparing solutionsof the delay and PDE formulations of a physiologically structured population modelwith distributed states at birth. We also note that the results established in Section4 can readily be extended (at least) for semilinear equations, when solutions of thePDE can be obtained using the method of characteristics. One anticipates that theresults carry forward from the linear case, but the calculations and formulas will bemuch more cumbersome. We anticipate though, that a similar comparison of other classesof models, in particular quasi-linear ones, may reveal differences. A natural candidatewould be a quasilinear hierarchic (infinite dimensional environment/interaction variable)size-structured population model with single state at birth. It was shown in [3, 4] thatthis model is not well-posed on the biologically natural state space L1, if the growthrate is not a monotone function of the environment. Specifically, in [4] it was shownthat finite time blow up of the density function is possible, and consequently, one hasto work with measures to study existence of solutions. At the same time note thatmodels of this type (for example the one studied in [4]) can be formulated and studied in


the delay equation framework, too. In particular, an interesting cannibalistic model withinfinite dimensional environment was shown to be well-posed in [16] in the delay equationframework, under the assumptions that the vital rates are continuously differentiable.

Motivated by the results obtained in Section 4 for the distributed states at birth model(2.1), we established some very general results in Section 5. Our viewpoint (and startingpoint) in Section 5 was also slightly different to that in Section 4. In particular, to allowgreater generality, and at the same time to avoid the delicate technical calculations ofSection 4, our starting point in Section 5 is the semigroup governing a PDE model. Thisthen allowed us to establish results in Section 5, which directly apply to the distributedstates at birth model (2.1). In particular, Theorem 5.16 establishes the equivalence of thesteady state problems in the two different formulations; while Theorem 5.13 establishesthe rigorous connection between time-dependent solutions of the two models.

However, to extend the results of Section 5 for nonlinear models, promises to berather challenging. There are also natural, biologically motivated examples of recruitmentprocesses, which lead to an unbounded operator C, posing additional difficulties. We aimto work on these problems in the future.

One may also ask the very natural question what happens in a perhaps simpler sin-gle state at birth model. (Note that when we say ”simpler” we mean from the mod-elling/biological point of view.) Indeed it will be very interesting to explore whetherTheorem 2.1 could be generalised, and one could use a similar operator splitting in caseof the single state at birth model, by considering the recruitment at state 0 as a boundaryperturbation. In fact there are numerous results, see e.g. [13, 21, 30], on boundary per-turbations of strongly continuous semigroups. The main idea of the Desch-Schappacherboundary perturbation theory is to lift the problem to the extrapolated space where onedefines an additive, bounded and positive perturbation. Then, the boundary pertur-bation, i.e. the recruitment operator, can be recovered as the part of the generator ofthe semigroup (defined on the extrapolated space) in the original state space. We mayanticipate that one will then apply a similar result to Theorem 2.1, for the generatorof the extrapolated semigroup. Since the extrapolated semigroup and the original semi-group are similar, their spectrum, spectral bound and growth bound coincide. We aimto elaborate the details of this in the future.

Acknowledgments

A. Calsina was partially supported by the research projects 2009SGR-345 and DGIMTM2011-27739-C04-02. J. Z. Farkas was supported by the research project DGI MTM2011-27739-C04-02, while visiting the Universitat Autonoma de Barcelona, and by a personalresearch grant from The Carnegie Trust for the Universities of Scotland.

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http://webarchive.iiasa.ac.at/Research/EEP/Metz2Book.html


Angel Calsina, Department of Mathematics, Universitat Autonoma de Barcelona, Bel-

laterra, 08193, Spain

E-mail address: [email protected]

Odo Diekmann, Department of Mathematics, University of Utrecht, Budapestlaan 6, PO

Box 80010, 3508 TA, Utrecht, The Netherlands


Jozsef Z. Farkas, Division of Computing Science and Mathematics, University of Stirling,

Stirling, FK9 4LA, United Kingdom


Documents

arXiv:1510.08624v2 [math.AP] 31 Jan 2016 · linear age-structured population dynamics; while [12] provides valuable insight into the modelling and analysis of size-structured populations,